Showing papers on "Cartan matrix published in 2022"
TL;DR: In this paper , the authors consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebra that are fixed by the Chevalley involution and study how these representations relate to spinor representations as they arise in the theory of supergravity.
Abstract: We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of $\mathbb{N}$-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity.
3 citations
TL;DR: The (q, t)-Cartan matrix has deep connections with the representation theory of its untwisted quantum affine algebra, and quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetric type as mentioned in this paper .
Abstract: The (q, t)-Cartan matrix specialized at $$t=1$$ , usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetric type. In this paper, we study the (q, t)-Cartan matrix specialized at $$q=1$$ , called the t-quantized Cartan matrix, and investigate the relations with (ii $$'$$ ) its corresponding unipotent quantum coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.
1 citations
TL;DR: In this paper , a natural notion of determinant in matrix JB$^*$-algebras was introduced, i.e., for hermitian matrices of biquaternions, and for their determinants were established several properties which are useful to understand the structure of the Cartan factor of type $6.
Abstract: We introduce a natural notion of determinant in matrix JB$^*$-algebras, i.e., for hermitian matrices of biquaternions and for hermitian $3\times 3$ matrices of complex octonions. We establish several properties of these determinants which are useful to understand the structure of the Cartan factor of type $6$. As a tool we provide an explicit description of minimal projections in the Cartan factor of type $6$ and a variety of its automorphisms.
1 citations
09 Mar 2022
TL;DR: In this paper , the geometry of Cartan-Hartogs has been studied from a symplectic point of view, inspired by duality between compact and non-compact Hermitian symmetric spaces.
Abstract: This paper studies the geometry of Cartan-Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan-Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan-Hartogs and their dual. Further, we compute their symplectic capacity and show that a Cartan-Hartogs admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.
06 May 2022
TL;DR: In this article , Cartan-theoretically proved exceptionality of the $3:1$ ratio for two 2-spheres rolling on each other without twisting or slipping, yielding a $(2,3,5)$-distribution with symmetry the Lie algebra of the split real form of $G_2.
Abstract: In his 1910 paper, \'Elie Cartan gave a tour-de-force solution to the (local) equivalence problem for generic rank 2 distributions on 5-manifolds, i.e. $(2,3,5)$-distributions. From a modern perspective, these structures admit equivalent descriptions as (regular, normal) parabolic geometries modelled on a quotient of $G_2$, but this is not transparent from his article: indeed, the Cartan "connection" of 1910 is not a "Cartan connection" in the modern sense. We revisit the classification of multiply-transitive $(2,3,5)$-distributions from a modern Cartan-geometric perspective, incorporating $G_2$ structure theory throughout, obtaining: (i) the complete (local) classifications in the complex and real settings, phrased "Cartan-theoretically", and (ii) the full curvature and infinitesimal holonomy of all these models. Moreover, we Cartan-theoretically prove exceptionality of the $3:1$ ratio for two 2-spheres rolling on each other without twisting or slipping, yielding a $(2,3,5)$-distribution with symmetry the Lie algebra of the split real form of $G_2$.
TL;DR: In this paper , the authors studied the graded identities for Kac-Moody algebras when the matrix A is diagonal and provided a vector space basis of the relatively free algebra.
TL;DR: In this paper , a family of quiver Hecke algebras which give a categorisation of quantum Borcherds algebra associated to an arbitrary Borcherd-Cartan datum is introduced.
18 May 2022
TL;DR: In this paper , a family of quiver Hecke algebras which give a categorisation of quantum Borcherds algebra associated to an arbitrary Borcherd-Cartan datum is introduced.
Abstract: We introduce a family of quiver Hecke algebras which give a categorification of quantum Borcherds algebra associated to an arbitrary Borcherds-Cartan datum.
07 Dec 2022
TL;DR: In this paper , the instanton partition function of quiver theory on various manifolds is computed using the $q$-Cartan matrix and the notion of double quiver gauge theory characterized by a pair of quivers.
Abstract: We provide a formalism using the $q$-Cartan matrix to compute the instanton partition function of quiver gauge theory on various manifolds. Applying this formalism to eight dimensional setups, we introduce the notion of double quiver gauge theory characterized by a pair of quivers. We also explore the BPS/CFT correspondence in eight dimensions based on the $q$-Cartan matrix formalism.
TL;DR: In this article, the necessary and sufficient conditions for which the tensor product of irreducible representations from O representations of Borcherds-Kac-Moody algebras over complex numbers is isomorphic to another.
12 Jul 2022
TL;DR: In this paper , a quantum loop group associated to a general symmetric Cartan matrix is introduced, which is used to describe the localized K-theoretic Hall algebra of any quiver without loops.
Abstract: We introduce a quantum loop group associated to a general symmetric Cartan matrix, by imposing just enough relations between the usual generators $\{e_{i,k}, f_{i,k}\}_{i \in I, k \in \mathbb{Z}}$ in order for the natural Hopf pairing between the positive and negative halves of the quantum loop group to be perfect. As an application, we describe the localized K-theoretic Hall algebra of any quiver without loops, endowed with a particularly important $\mathbb{C}^*$ action.
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07 Feb 2022TL;DR: In this article , the authors define the outer automorphism group of a real form of a contragredient Lie superalgebra, and express Out(g) in terms of diagram symmetries on the Kac diagrams and Cartan automorphisms.
TL;DR: In this paper , a linear combination of Cartan frame fields with non-constant differentiable functions is used to select the generating lines of the surfaces with null Cartan base curve.
Abstract: In this paper, some special types of surfaces with null Cartan base curve are investigated. The generating lines of the surfaces are chosen as a linear combination of Cartan frame fields with non-constant differentiable functions. Firstly, the surfaces whose generating lines have the same direction of Cartan frame fields B; N and T are examined respectively. As a special case, Gaussian and Mean curvatures of one parameter family of Bertrand curves of a given null Cartan curve and the singular points of this type of surface are stated. Furthermore, an example is also stated to explain the obtained results. Then, the surfaces with null Cartan base curve are investigated where generating lines lie on the planes spanned by (N, B), (T,B) and (T, N) respectively. Finally, some differential geometric properties of these surface are given mainly in three different cases
27 Jan 2022
TL;DR: In this article , the authors studied the quantum Cartan matrix specialized at $q = 1 and investigated the relations with quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.
Abstract: The $(q,t)$-Cartan matrix specialized at $t=1$, usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of kew-symmetric type. In this paper, we study the $(q,t)$-Cartan matrix specialized at $q=1$, called the $t$-quantized Cartan matrix, and investigate the relations with (ii') its corresponding quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.
20 Jun 2022
TL;DR: In this article , essential commutative Cartan pairs of Ω(C^*$-algebras generalising the definition of Renault and showing that such pairs are given by essential twisted groupoid is defined by Kwa\'sniewski and Meyer.
Abstract: We define essential commutative Cartan pairs of $C^*$-algebras generalising the definition of Renault and show that such pairs are given by essential twisted groupoid $C^*$-algebras as defined by Kwa\'sniewski and Meyer. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential commutative Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions.